Discrepancy of lattice points in round convex bodies by Gyula Karolyi Consider a sequence P of N points and a family F of measurable test sets in the unit square. For an element A of F, the discrepancy of P with respect to A is the difference between the "actual number" of points in A (that is, |P\cap A|) and the "expected number" of such points, which is N times the area of A. Taking the supremum over all test sets we get the discrepancy of P with respect to F. According to the Schmidt-Halasz-Wagner theorem, in case of axis-parallel rectangles the discrepancy is always at least clogN for some positive constant c. Moreover, explicit number theoretic constructions show that this lower bound cannot be improved upon. Note that the discrepancy of a random point set would be roughly \sqrt{N}. Due to the work of Beck, Matousek, Roth, Schmidt, Welzl, and more, similar results exist for various families F. However, apart from the above example, no explicit constructions are known that match the lower bounds. For example, with respect to circles in the unit square, every N-element sequence has a discrepancy at least N^{1/4}, but this order of magnitude can only be justified by the probabilistic method. With respect to all convex regions the discrepancy jumps up to roughly N^{1/3}. For a fixed constant r>1, the convex region A with smooth (C^2) boundary is called r-round, if the ratio between the largest and smallest curvature around the boundary does not exceed r. Let F=F(r) be the family of all r-round convex regions in the unit square. Based on a beautiful combinatorial idea of Schmidt we prove that every N-sequence has a discrepancy at least c(r)N^{1/3} with respect to F(r). Using a result of Barany and Larman we also show that appropriately scaled grids meet this lower bound.